Analytical and numerical methods for spillover effects in prioritized PrEP for HIV prevention

Pre-exposure prophylaxis (PrEP) is an effective intervention for preventing HIV transmission, but high cost and uneven uptake raise challenges for resource allocation. While spillover effects, wherein PrEP use in one group reduces infections in others, are known to occur, they remain poorly quantified and rarely guide policy. We provide a comprehensive modeling study for PrEP spillover across risk groups, and develop analytic and numerical tools for its quantification. We first develop a compartmental model for HIV transmission that stratifies the population into interacting subgroups: heterosexual males (HETM), high- and low-risk heterosexual females (HETF-hi/HETF-lo) and men who have sex with men (MSM). The asymptotic stability of the disease-free equilibrium of the model is analyzed. Spillover is quantified by deriving an expression for the spillover-adjusted number needed to treat (NNT), a measure of the population-level impact of PrEP. Simulations show PrEP delivery to MSM yields substantial indirect benefits, particularly for HETF-lo, where spillover exceeds the direct effect by a factor of five. We show targeting HETF-hi outperforms direct PrEP delivery to HETM, emphasizing the importance of intra-group heterogeneity. To evaluate whether these results hold under more detailed assumptions, we embed our framework into the national HOPE model maintained by the Centers for Disease Control and Prevention (CDC) and conduct global sensitivity analysis using Sobol indices with Polynomial Chaos Expansion. This approach extends our analytical insights and quantifies how uncertainty in PrEP allocation propagates through complex dynamics. Further, this framework provides a numerical procedure for quantifying spillover where direct analysis is impractical. Our results show that spillover is a central driver of PrEP dynamics and that failing to account for it risks mis-allocating resources.